Kinetic energy

Moving things can do work

A moving ball knocks down a row of blocks; a fast stone breaks a window, and a slow one does not. A truck is more dangerous than a cycle at the same speed. So the energy of a moving object depends on its speed and its mass. This energy of motion is its kinetic energy.

How much kinetic energy?

Push an object of mass $m$, at rest, with a steady force $F$ over a distance $s$. It reaches a speed $v$.

1. The work done is $W = F \times s$, and Newton's second law gives $F = ma$.
2. From rest, $v^2 = 2as$, so $s = \dfrac{v^2}{2a}$.
3. So $W = ma \times \dfrac{v^2}{2a} = \tfrac{1}{2}mv^2$.

All this work is now in the object as kinetic energy:

with $m$ in kilograms, $v$ in metres per second, and $KE$ in joules.

The speed matters most

The speed is squared. Twice the speed gives $2^2 = 4$ times the energy; three times the speed gives $9$ times. Twice the mass only doubles it.

You should find that the hollow point, at twice the speed, always sits four times as high. That is why a fast car needs so much more road to stop.

Work changes the kinetic energy

The work done by the net force equals the change in kinetic energy:

Positive work speeds an object up. Negative work, like a brake's, slows it down.

Watch out. Square only the speed, and do not forget the half. $\tfrac{1}{2} \times 2\ \text{kg} \times (3\ \text{m/s})^2 = 9\ \text{J}$, not $2\ \text{kg} \times 3\ \text{m/s} = 6\ \text{J}$.

A worked example

A girl of $45\ \text{kg}$ runs at $6\ \text{m/s}$, then slows to $3\ \text{m/s}$. Find her kinetic energy at each speed.

1. At 6 m/s. $KE = \tfrac{1}{2} \times 45\ \text{kg} \times (6\ \text{m/s})^2 = 810\ \text{J}$.
2. At 3 m/s. $KE = \tfrac{1}{2} \times 45\ \text{kg} \times (3\ \text{m/s})^2 = 202.5\ \text{J}$.
3. So half the speed leaves a quarter of the energy. Her legs did $202.5\ \text{J} - 810\ \text{J} = -607.5\ \text{J}$ of work to slow her.

Practise with new numbers

Try it in Python

The program finds the kinetic energy of a car at different speeds.

Remember these

• $KE = \tfrac{1}{2}mv^2$, in joules.
• Twice the speed gives four times the kinetic energy; twice the mass gives twice.
• The work done on an object equals the change in its kinetic energy.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Find the kinetic energy of a $0.45\ \text{kg}$ football kicked at $12\ \text{m/s}$.

2. A cycle speeds up from $4\ \text{m/s}$ to $8\ \text{m/s}$. By how many times does its kinetic energy grow?

3. A car of $1000\ \text{kg}$ moves at $10\ \text{m/s}$. How much work must the brakes do to stop it?

4. Which has more kinetic energy: a $3\ \text{kg}$ ball at $2\ \text{m/s}$, or a $1\ \text{kg}$ ball at $4\ \text{m/s}$?